The study of triangular
AF algebras has focused mostly on the classification, and although some
canonical representations of the important examples have long been known,
there has been little systematic study of the representation theory of these
algebras. The representations considered here are those which are restrictions of
star-representations of the full AF algebra. Furthermore, attention will mostly be
restricted to representations which map the masa of the triangular AF algebra
weakly densely into a masa of B(H). Such representations have a convenient
description using groupoids. Much of the paper consists of examples illustrating what
can and cannot occur.
